Relationships between knowledge bases and their uncertainty measures
Introduction
Rough set theory, presented by Pawlak [12], is a mathematical tool to deal with uncertainty and can be considered as the generalization of classical set theory. It has been successfully applied to intelligent systems, expert systems, knowledge discovery, pattern recognition, machine learning, signal analysis, image processing, inductive reasoning, and decision analysis, and in many other fields [12], [13], [14], [15].
A knowledge base is a basic notion of rough set theory. One of the strengths of rough set theory is that an unknown target concept can be approximately characterized by existing knowledge structures in a knowledge base. For a given knowledge base, one of the tasks in data mining and knowledge discovery is to generate new knowledge through use of known knowledge. From the viewpoint of rough set theory, knowledge means the capacity to classify objects [3], [7], [12], where objects may be real things, abstract concepts, processes, states, and moments of time. Initially, rough set theory just considers equivalence relations. Every equivalence relation on a given universe determines a classification on this universe and vice versa. One deals with not only a single classification on the universe but also a family of classifications on the universe [3], [8]. This leads to the notion of knowledge bases.
The study of knowledge bases is an important research issue. Some scholars have done very good work. For example, Qian et al. [17] studied knowledge structure in a tolerance knowledge base by means of set families and proposed knowledge distance between knowledge structures. Li et al. [5] discussed relationships between knowledge bases and proved that knowledge reductions, coordinate families and necessary relations in a knowledge base are invariant and inverse invariant under homomorphisms. Li et al. [6] studied properties of knowledge structures in a knowledge base by means of condition information amounts, inclusion degrees, and lower approximation operators, and gave group, lattice, mapping, and soft characterizations of knowledge structures in a knowledge base. Qin [16] showed that ⁎-reductions in a knowledge base are invariant and inverse invariant under homomorphisms.
The concept of entropy originates from energetics. It can be used to measure the degree of disorder of a system. The entropy of a system, proposed by Shannon [18], gives a measure of the uncertainty of a system. It has been applied in diverse fields as a useful mechanism for evaluating uncertainty in various modes. Some scholars have applied the extension of entropy and its variants to rough sets. For example, Düntsch and Gediga [2] proposed information entropy and three kinds of conditional entropy in rough sets for prediction of a decision attribute. Beaubouef et al. [1] presented a method measuring uncertainty of rough sets and rough relation databases. Wierman [19] addressed granulation measure to measure uncertainty of information. Yao [23] gave a granularity measure from the angle of granulation. Liang et al. [9], [11] studied several measures of knowledge in incomplete and complete information systems. Liang and Shi [10] introduced information entropy, rough entropy, and knowledge granulation in rough set theory; Qian et al. [17] investigated knowledge granulation of knowledge structures in a tolerance knowledge base.
The aim of this article is to investigate relationships between knowledge bases and their uncertainty measures. We perform statistical analysis of the proposed measures to determine their effectiveness or merits in a statistical sense.
The rest of this article is organized as follows. In Section 2, some basic notions of knowledge bases, relation information systems, and homomorphisms between relation information systems are recalled. In Section 3, dependence and independence between knowledge bases are proposed, and knowledge distance between knowledge bases is introduced. In Section 4, invariant and inverse invariant characteristics of knowledge bases under homomorphisms based on data compression are obtained. In Section 5, some tools for measuring uncertainty of knowledge bases are introduced, and an illustrative example is given. In Section 6, a numerical experiment is given, and effectiveness analysis is conducted from the viewpoint of statistics. Section 7 provides a summary.
Section snippets
Preliminaries
In this section, we recall some basic notions of knowledge bases, relation information systems, and homomorphisms.
Throughout this article, denotes two nonempty finite sets called the universes, denotes the family of all subsets of U, and denotes the cardinality of . All mappings are assumed to be surjective.
Let
Relationships between knowledge bases
In this section, we investigate relationships between knowledge bases from the two aspects of dependence and separation.
Invariant and inverse invariant characteristics of knowledge bases under homomorphisms
In this section, invariant and inverse invariant characteristics of knowledge bases under homomorphisms are obtained.
Theorem 4.1 Let . Suppose . If , , then
Proof “⟹.” Suppose . Then for all i, . Let . For all , we have ; that is, . By Theorem 2.21, Then ; that is, . Thus there exists , , , and .
Some tools for measuring uncertainty of knowledge bases
In this section, we investigate measuring uncertainty of knowledge bases.
Numerical experiments and effectiveness analysis
In this section, we conduct a numerical experiment and perform effectiveness analysis from the two aspects of dispersion and correlation in statistics.
Conclusions
In this article, relationships between knowledge bases have been studied from the two aspects of dependence and separation. Measurement of uncertainty of knowledge bases was investigated, and from the viewpoint of statistics, effectiveness analysis was conducted by a numerical experiment. Invariant and inverse invariant characteristics of knowledge bases under homomorphisms based on data compression were obtained (see Table 6). These results will be important in building a framework of granular
Acknowledgements
The authors thank the editors and the anonymous reviewers for their valuable suggestions that helped immensely in improving the quality of this article. This work was supported by the National Natural Science Foundation of China (11461005), the Natural Science Foundation of Guangxi Province (2016GXNSFAA380045, 2016GXNSFAA380282, 2016GXNSFAA380286), the National Social Science Foundation of China (12BJL087), and the Philosophy and Social Sciences Planning Project of Guangxi (11BJY029).
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